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Average behavior of Fourier coefficients of cusp forms

Author: Guangshi Lü
Journal: Proc. Amer. Math. Soc. 137 (2009), 1961-1969
MSC (2000): Primary 11F30, 11F11, 11F66
Published electronically: December 30, 2008
MathSciNet review: 2480277
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Abstract: Let $ a_0(n)$ and $ b_0(n)$ be the normalized Fourier coefficients of the two holomorphic Hecke eigenforms $ f(z)\in S_{2k}(\Gamma)$ and $ \varphi(z)\in S_{2l}(\Gamma)$ respectively. In 1999, Fomenko studied the following average sums of $ a_0(n)$ and $ b_0(n)$:

$\displaystyle \sum_{n \leq x}a_0(n)^3, \quad \sum_{n \leq x}a_0(n)^2b_0(n), \quad \sum_{n \leq x}a_0(n)^2b_0(n)^2, \quad \sum_{n \leq x}a_0(n)^4.$      

In this paper, we are able to improve on Fomenko's results.

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Additional Information

Guangshi Lü
Affiliation: Department of Mathematics, Shandong University, Jinan, Shandong, 250100, People’s Republic of China

Keywords: Fourier coefficients of cusp forms, Gelbart-Jacquet lift, $L$-function
Received by editor(s): May 30, 2008
Received by editor(s) in revised form: August 28, 2008
Published electronically: December 30, 2008
Additional Notes: This work was supported by the National Natural Science Foundation of China (Grant No. 10701048).
Communicated by: Ken Ono
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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